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The Markoff Automaton: A New Algorithm For Simulating The Time-Evolution Of Large Stochastic Dynamic Systems

Author

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  • THOMAS FRICKE

    (Institut für Physik, Humboldt-Universität, Invalidenstr. 110, D-100 99 Berlin, Germany)

  • DIETMAR WENDT

    (Institut für Theoretische Physik, Rheinisch Westfälische Technische Hochschule Aachen, Sommerfeldstraβe, D-52 056 Aachen, Germany)

Abstract

We describe a new algorithm for simulating complex Markoff processes. We have used a reaction-cell method in order to simulate arbitrary reactions. It can be used for any kind of RDS on arbitrary topologies, including fractal dimensions or configurations not being related to any spatial geometry. The events within a single cell are managed by an event handler which has been implemented independently of the system studied. The method is exact on the Markoff level including the correct treatment of finite numbers of molecules.To demonstrate its properties, we apply it on a very simple reaction-diffusion-systems (RDS). The chemical equations$A+A\to\emptyset$and$A+B\to\emptyset$in 1 to 4 dimensions serve as models for systems whose dynamics on an intermediate time scale are governed by fluctuations. We compare our results to the analytic approach by the scaling ansatz. The simulations confirm the exponents of theA+Bsystem within statistical errors, including the logarithmic corrections in the dimensiond=2.The method is able to simulate the crossover from the reaction to diffusion limited regime, which is defined be a crossover time depending on the system size.

Suggested Citation

  • Thomas Fricke & Dietmar Wendt, 1995. "The Markoff Automaton: A New Algorithm For Simulating The Time-Evolution Of Large Stochastic Dynamic Systems," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 6(02), pages 277-306.
    • Handle: RePEc:wsi:ijmpcx:v:06:y:1995:i:02:n:s0129183195000216
    DOI: 10.1142/S0129183195000216
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    Cited by:

    1. Jaekel, Uwe, 2005. "Pricing of American style options with an adjoint process correction method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 352(2), pages 584-600.
    2. Justin S Hogg & Leonard A Harris & Lori J Stover & Niketh S Nair & James R Faeder, 2014. "Exact Hybrid Particle/Population Simulation of Rule-Based Models of Biochemical Systems," PLOS Computational Biology, Public Library of Science, vol. 10(4), pages 1-16, April.

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